Global attractors for the [formula omitted]-Laplacian equations with nonregular data

Abstract

Let Ω be a smooth bounded domain in RN,(N≥3). We consider the long time behavior of solutions to the p-Laplacian equation ut−Δpu+f(u)=g where 2≤p<N,−Δp=−div(|∇u|p−2∇u) is the p-Laplace operator. Assume that g and the initial condition u0 lie in L1(Ω), f:R→R is of class C1 and satisfies proper growth conditions. Firstly, we prove the uniqueness of the entropy solution and establish some regularity results. Then we show the existence of a global attractor A in Lr−1(Ω)∩W01,s(Ω) with s<max{N(p−1)N−1,p(r−1)r}. To obtain the results, a decomposition method and a bootstrap technique are used.